Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Extreme Value Laws for non stationary processes generated by sequential and random dynamical systems

Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Sandro Vaienti

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We develop and generalise the theory of extreme value for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. We apply our results to non-autonomous dynamical systems, in particular to sequential dynamical systems, given by uniformly expanding maps, and to a few classes of random dynamical systems. Some examples are presented and worked out in detail.


Nous développons et généralisons la théorie des valeurs extrêmes pour des processus stochastiques non-stationnaires, en affaiblissant la condition de mélange uniforme qui avait été utilisée auparavant. Nous appliquons nos résultats à des systèmes dynamiques non autonomes, en particulier aux systèmes dynamiques séquentiels engendrés par des applications dilatantes et à une large classe de systèmes dynamiques aléatoires. Quelques exemples sont présentés et calculés en détail.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1341-1370.

Received: 1 December 2015
Revised: 4 April 2016
Accepted: 6 April 2016
First available in Project Euclid: 21 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 60G70: Extreme value theory; extremal processes 37B20: Notions of recurrence 37A25: Ergodicity, mixing, rates of mixing

Non-stationarity Extreme value theory Hitting Times Sequential dynamical systems Random dynamical systems


Freitas, Ana Cristina Moreira; Freitas, Jorge Milhazes; Vaienti, Sandro. Extreme Value Laws for non stationary processes generated by sequential and random dynamical systems. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1341--1370. doi:10.1214/16-AIHP757.

Export citation


  • [1] R. Aimino, H. Hu, M. Nicol, A. Török and S. Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete Contin. Dyn. Syst. 35 (3) (2015) 793–806. doi:10.3934/dcds.2015.35.793.
  • [2] J. F. Alves, J. M. Freitas, S. Luzzatto and S. Vaienti. From rates of mixing to recurrence times via large deviations. Adv. Math. 228 (2) (2011) 1203–1236. Available at arXiv:1007.3771. doi:10.1016/j.aim.2011.06.014.
  • [3] L. Arnold. Random Dynamical Systems. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi:10.1007/978-3-662-12878-7.
  • [4] H. Aytaç, J. M. Freitas and S. Vaienti. Laws of rare events for deterministic and random dynamical systems. Trans. Amer. Math. Soc. 367 (11) (2015) 8229–8278. doi:10.1090/S0002-9947-2014-06300-9.
  • [5] W. Bahsoun and S. Vaienti. Escape rates formulae and metastability for randomly perturbed maps. Nonlinearity 26 (5) (2013) 1415–1438. doi:10.1088/0951-7715/26/5/1415.
  • [6] D. Berend and V. Bergelson. Ergodic and mixing sequences of transformations. Ergodic Theory Dynam. Systems 4 (3) (1984) 353–366. doi:10.1017/S0143385700002509.
  • [7] M. R. Chernick, T. Hsing and W. P. McCormick. Calculating the extremal index for a class of stationary sequences. Adv. in Appl. Probab. 23 (4) (1991) 835–850. doi:10.2307/1427679.
  • [8] P. Collet. Statistics of closest return for some non-uniformly hyperbolic systems. Ergodic Theory Dynam. Systems 21 (2) (2001) 401–420. doi:10.1017/S0143385701001201.
  • [9] J.-P. Conze and A. Raugi. Limit theorems for sequential expanding dynamical systems on $[0,1]$. In Ergodic Theory and Related Fields 89–121. Contemp. Math. 430. Amer. Math. Soc., Providence, RI, 2007. doi:10.1090/conm/430/08253.
  • [10] M. Falk, J. Hüsler and R.-D. Reiss. Laws of Small Numbers: Extremes and Rare Events, extended edition. Birkhäuser/Springer Basel AG, Basel, 2011. doi:10.1007/978-3-0348-0009-9.
  • [11] A. C. M. Freitas and J. M. Freitas. On the link between dependence and independence in extreme value theory for dynamical systems. Statist. Probab. Lett. 78 (9) (2008) 1088–1093. doi:10.1016/j.spl.2007.11.002.
  • [12] A. C. M. Freitas, J. M. Freitas and M. Todd. Hitting time statistics and extreme value theory. Probab. Theory Related Fields 147 (3–4) (2010) 675–710. Available at arXiv:0804.2887. doi:10.1007/s00440-009-0221-y.
  • [13] A. C. M. Freitas, J. M. Freitas and M. Todd. Extreme value laws in dynamical systems for non-smooth observations. J. Stat. Phys. 142 (1) (2011) 108–126. Available at arXiv:1006.3276. doi:10.1007/s10955-010-0096-4.
  • [14] A. C. M. Freitas, J. M. Freitas and M. Todd. The extremal index, hitting time statistics and periodicity. Adv. Math. 231 (5) (2012) 2626–2665. Available at arXiv:1008.1350. doi:10.1016/j.aim.2012.07.029.
  • [15] A. C. M. Freitas, J. M. Freitas and M. Todd. Speed of convergence for laws of rare events and escape rates. Stochastic Process. Appl. 125 (4) (2015) 1653–1687. Available at arXiv:1401.4206. doi:10.1016/
  • [16] A. C. M. Freitas, J. M. Freitas and S. Vaienti Extreme value laws for non stationary, sequential intermittent systems. Preprint, 2016. Available at arXiv:1605.06287.
  • [17] C. González-Tokman, B. R. Hunt and P. Wright. Approximating invariant densities of metastable systems. Ergodic Theory Dynam. Systems 31 (5) (2011) 1345–1361. doi:10.1017/S0143385710000337.
  • [18] N. Haydn, M. Nicol, A. Török and S. Vaienti Almost sure invariance principle for sequential and non-stationary dynamical systems. Preprint. Available at arXiv:1406.4266.
  • [19] N. Haydn, M. Nicol, S. Vaienti and L. Zhang. Central limit theorems for the shrinking target problem. J. Stat. Phys. 153 (5) (2013) 864–887. doi:10.1007/s10955-013-0860-3.
  • [20] H. Hu and S. Vaienti. Absolutely continuous invariant measures for non-uniformly expanding maps. Ergodic Theory Dynam. Systems 29 (4) (2009) 1185–1215. doi:10.1017/S0143385708000576.
  • [21] J. Hüsler. Asymptotic approximation of crossing probabilities of random sequences. Z. Wahrsch. Verw. Gebiete 63 (2) (1983) 257–270. doi:10.1007/BF00538965.
  • [22] J. Hüsler. Extreme values of nonstationary random sequences. J. Appl. Probab. 23 (4) (1986) 937–950.
  • [23] Y. Kifer. Ergodic Theory of Random Transformations. Progress in Probability and Statistics 10. Birkhäuser Boston, Inc., Boston, MA, 1986. doi:10.1007/978-1-4684-9175-3.
  • [24] Y. Kifer. Random Perturbations of Dynamical Systems. Progress in Probability and Statistics 16. Birkhäuser Boston, Inc., Boston, MA, 1988. doi:10.1007/978-1-4615-8181-9.
  • [25] Y. Kifer. Limit theorems for random transformations and processes in random environments. Trans. Amer. Math. Soc. 350 (4) (1998) 1481–1518. doi:10.1090/S0002-9947-98-02068-6.
  • [26] Y. Kifer and A. Rapaport. Poisson and compound Poisson approximations in conventional and nonconventional setups. Probab. Theory Related Fields 160 (3–4) (2014) 797–831. doi:10.1007/s00440-013-0541-9.
  • [27] M. Nicol, A. Török and S. Vaienti Central limit theorems for sequential and random intermittent dynamical systems. Preprint, 2015. Available at arXiv:1510.03214.
  • [28] X.-F. Niu. Extreme value theory for a class of nonstationary time series with applications. Ann. Appl. Probab. 7 (2) (1997) 508–522. doi:10.1214/aoap/1034625342.
  • [29] W. Parry. On the $\beta$-expansions of real numbers. Acta Math. Acad. Sci. Hung. 11 (1960) 401–416.
  • [30] J. Rousseau. Hitting time statistics for observations of dynamical systems. Nonlinearity 27 (9) (2014) 2377–2392. doi:10.1088/0951-7715/27/9/2377.
  • [31] J. Rousseau, B. Saussol and P. Varandas. Exponential law for random subshifts of finite type. Stochastic Process. Appl. 124 (10) (2014) 3260–3276. doi:10.1016/
  • [32] J. Rousseau and M. Todd. Hitting times and periodicity in random dynamics. J. Stat. Phys. 161 (1) (2015) 131–150. doi:10.1007/s10955-015-1325-7.
  • [33] B. Saussol. Absolutely continuous invariant measures for multidimensional expanding maps. Israel J. Math. 116 (2000) 223–248. doi:10.1007/BF02773219.